Hamilton Formalism in Non-Commutative Geometry

TL;DR

This paper develops a Hamilton formalism for Yang-Mills theory within non-commutative geometry, enabling the definition of phase space and Poisson brackets in Minkowskian space-time, and applies it to specific models.

Contribution

It introduces a Hamiltonian framework for non-commutative geometric models, including Minkowskian space-time, with a novel approach to defining integration and phase space.

Findings

Defined non-commutative phase space and Poisson brackets for Yang-Mills theory.

Applied the formalism to models on two-point space and Minkowski space-time.

Extended Hamiltonian methods to non-commutative geometrical settings.

Abstract

We study the Hamilton formalism for Connes-Lott models, i.e., for Yang-Mills theory in non-commutative geometry. The starting point is an associative $*$-algebra $\cA$ which is of the form $\cA=C(I,\cAs)$ where $\cAs$ is itself a associative $*$-algebra. With an appropriate choice of a k-cycle over $\cA$ it is possible to identify the time-like part of the generalized differential algebra constructed out of $\cA$. We define the non-commutative analogue of integration on space-like surfaces via the Dixmier trace restricted to the representation of the space-like part $\cAs$ of the algebra. Due to this restriction it possible to define the Lagrange function resp. Hamilton function also for Minkowskian space-time. We identify the phase-space and give a definition of the Poisson bracket for Yang-Mills theory in non-commutative geometry. This general formalism is applied to a model on a two-point space and to a model on Minkowski space-time $\times$ two-point space.